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\title{REPORT}
\author{Daniel Zawada}
\date{June 23nd 2011}


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Consider a particle undergoing brownian motion in a unit sphere denoted by $\Omega$, the boundary of the sphere ($\partial \Omega$)
is made up of completely reflecting patches, $\partial \Omega_r$ and completely absorbing patches, $\partial \Omega_a$.  The mean first
passage time, $v(x)$, is defined as the expectation value of the time for a particle starting at $x$ to be absorbed on the boundary.
It is known that $v(x)$ satisfies,

\begin{equation}
	\begin{split}
		&\Delta v = -\frac{1}{D}, \hspace{5mm} x \in \Omega \\
		& v= 0, \hspace{5mm} x \in \partial \Omega_a \hspace{5mm} \partial_n v = 0,\hspace{5mm} x \in \partial \Omega_r
    \end{split}
\end{equation}

Define the average MFPT as, $\bar{v} = \int v(x) dx$ where the integral is over the domain $\Omega$. 
For $N$ traps, $\bar{v}$ was calculated asymptotically to be,

\begin{equation}
	\label{eqn:vbarasympt}
	\bar{v} = \frac{|\Omega|}{4 \epsilon D N} \bigg[1 + \frac{\epsilon}{\pi} \log\bigg(\frac{2}{\epsilon}\bigg) + \frac{\epsilon}{\pi}\bigg(-\frac{9 N}{5} + 2(N-2)\log 2 + \frac{3}{2} + \frac{4}{N} \mathcal{H}(x_1,\dots,x_N)\bigg)\bigg] 
\end{equation}

Where $x_i$'s are the trap locations with radius $\epsilon$ and $\mathcal{H}(x_1,\dots,x_N)$ is the interaction energy defined by,

\begin{equation}
	\label{eqn:hdef}
	\mathcal{H}(x_1,\dots,x_N) = \sum_{i=1}^N \sum_{j=i+1}^N \bigg(\frac{1}{|x_i-x_j|} - \frac{1}{2} \log |x_i-x_j| - \frac{1}{2} \log(2 + |x_i - x_j|)\bigg)
\end{equation}

We now attempt to evaluate the sum in (\ref{eqn:hdef}) in the limit $N \gg 1$ by approximating the sum as an integral.  Consider 
one trap located at $(r,\theta, \phi) = (1,0,0)$.  We assume the traps are distributed normally along the sphere except in a small
neighbourhood centered at the trap.  We write the trap density as,

\begin{equation}
	P(\theta,\phi) = \left\{\begin{array}{c c}
		0, & 0 < \theta < \theta_0 \\
		\frac{N}{4 \pi}, & \theta_0 < \theta < \pi 
	\end{array}\right.
\end{equation}

$\theta_0$ can be determined from the condition,

\begin{equation}
	\int_0^{2 \pi} \int_{\theta_0}^\pi P(\theta,\phi) \sin \theta d\theta d\phi = N-1
\end{equation}

Evaluating this integral yields $\cos \theta_0 = 1 - 2/N$.  Now we can approximate the sum as,

\begin{equation}
	\mathcal{H} \approx \frac{N}{2} \int_0^{2 \pi} \int_{\theta_0}^\pi P(\theta,\phi) \bigg(\frac{1}{r(\theta)} - \frac{1}{2} \log r(\theta) - \frac{1}{2} \log (2 + r(\theta))\bigg) \sin \theta d\theta d\phi
\end{equation}

Where we have multiplied by $1/2$ since the interactions are counted twice.  This integral yields,

\begin{equation}
	\label{eqn:Hintegral}
	\begin{split}
		\mathcal{H} \approx & N^2 \frac{1}{2} (1-\log 2) - \frac{1}{4} N^2 \log(1+\sqrt{N}) + \frac{1}{8} N^2 \log N - \frac{1}{4} N^{3/2} \\
	& - \frac{1}{4} N \log N +\frac{1}{4} N \log (\sqrt{N} + 1) - \frac{N}{2} \bigg(\frac{1}{2} - \log 2\bigg)
	\end{split}
\end{equation}

It is assumed that the leading order coefficient is correct since it is independant of the choice of $\theta_0$, we take the first
two terms of the series expansion for $\mathcal{H} \approx N^2 /2 \cdot (1-\log 2) + b_1 N^{3/2}$ and apply this to (\ref{eqn:vbarasympt})

\begin{equation}
	\bar{v} = \frac{|\Omega|}{4 \epsilon D N} \bigg[1-\frac{\epsilon}{\pi} \log \epsilon + \frac{\epsilon N}{\pi}\bigg(\frac{1}{5} + \frac{4 b_1}{\sqrt{N}}\bigg)\bigg] 
\end{equation}

For this equation to be valid the terms have to remain ordered, i.e.

\begin{equation}
	1 \gg -\frac{\epsilon}{\pi} \log \epsilon \gg \frac{\epsilon N}{\pi}\bigg(\frac{1}{5} + \frac{4 b_1}{\sqrt{N}}\bigg) 
\end{equation}

To satisfy this choose $\epsilon = \exp(-N^\alpha)$, then we have

\begin{equation}
	1 \gg \frac{e^{-N^\alpha} N^\alpha}{\pi} \gg \frac{e^{-N^\alpha} N}{\pi} \bigg(\frac{1}{5} + \frac{4 b_1}{\sqrt{N}}\bigg)
\end{equation}

For $N>1$ and $\alpha$ large both ordering conditions hold.  We can also calculate $\bar{v}$ through homegenization theory, where we replace
the mixed boundary conditions with robin boundary conditions. The problem becomes,

\begin{equation}
	\Delta v_H = \frac{-1}{D}, \hspace{5mm} v_H \in \Omega ; \hspace{5mm} \epsilon \partial_r v_H + \kappa v_H = 0,\hspace{5mm} v_H \in \partial \Omega
\end{equation}

Where $v_H=v_H(r)$ is the MFPT, $\epsilon$ is the radius of each trap, and $\kappa$ is a geometric factor determined by the type of traps and their placement. This equation has solution,

\begin{equation}
	\label{eqn:vhomeg}
	v_H = \frac{-r^2}{6 D} + \frac{1}{6 D} + \frac{1}{3 D} \frac{\epsilon}{\kappa}
\end{equation}

Thus the average MFPT for $v_H$ given by (\ref{eqn:vhomeg}) is,

\begin{equation}
	\label{eqn:vbarhomeg}
	\bar{v}_H = \frac{1}{15 D} + \frac{1}{3 D} \frac{\epsilon}{\kappa} 
\end{equation}


We now equate $\bar{v}$ and $\bar{v}_H$ and solve for $\kappa$ in terms of $\sigma = N \epsilon^2 /4$ to obtain,

\begin{equation}
	\kappa = \frac{4\sigma}{\pi} - \frac{32 b_1 \sigma^{3/2}}{\pi^2} + \frac{256 b_1^2 \sigma^2}{\pi^3} - \cdots + \mathcal{O}(\sigma^{\alpha+1})
\end{equation}

Which is the series form of the function,

\begin{equation}
	\kappa = \frac{4 \sigma}{\pi + 8 b_1 \sqrt{\sigma}}
\end{equation}

For $\sigma < 0.01$ we can neglect the $\sqrt{\sigma}$ term and obtain $\kappa \approx 4 \sigma / \pi$ this was compared to results
for $\kappa$ obtained by curve fitting (\ref{eqn:vbarhomeg}) to (\ref{eqn:vbarasympt}) when numerically obtained results for $\mathcal{H}$ 
were used.  

\begin{figure}[h!ps]
	\includegraphics[scale=0.5]{kappawithoutb1}
\end{figure}

There is reasonable agreement for small $\sigma$,
We can also fit the equation for $\kappa$ including $b_1$ to the same numerical results to obtain an estimate for $b_1$,
\newpage

\begin{figure}[h!ps]
	\includegraphics[scale=0.5]{kappawithb1}
\end{figure}

which yields $b_1 = -0.3672$

\end{document}
